Is there an integral fusion category of rank $7$, FPdim $210$ and type $[[1,1],[5,3],[6,1],[7,2]]$, with the following fusion rules?

$\small{\begin{smallmatrix} 1 & 0 & 0 & 0& 0& 0& 0 \\ 0 & 1 & 0 & 0& 0& 0& 0 \\ 0 & 0 & 1 & 0& 0& 0& 0 \\ 0 & 0 & 0 & 1& 0& 0& 0 \\ 0 & 0 & 0 & 0& 1& 0& 0 \\ 0 & 0 & 0 & 0& 0& 1& 0 \\ 0 & 0 & 0 & 0& 0& 0& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 1 & 0 & 0& 0& 0& 0 \\ 1 & 1 & 0 & 1& 0& 1& 1 \\ 0 & 0 & 1 & 0& 1& 1& 1 \\ 0 & 1 & 0 & 0& 1& 1& 1 \\ 0 & 0 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 1 & 0& 0& 0& 0 \\ 0 & 0 & 1 & 0& 1& 1& 1 \\ 1 & 1 & 1 & 0& 0& 1& 1 \\ 0 & 0 & 0 & 1& 1& 1& 1 \\ 0 & 1 & 0 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 1& 0& 0& 0 \\ 0 & 1 & 0 & 0& 1& 1& 1 \\ 0 & 0 & 0 & 1& 1& 1& 1 \\ 1 & 0 & 1 & 1& 0& 1& 1 \\ 0 & 1 & 1 & 0& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 0& 1& 0& 0 \\ 0 & 0 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 0 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 0& 1& 1& 1 \\ 1 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 2& 1 \\ 0 & 1 & 1 & 1& 1& 1& 2 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 0& 0& 1& 0 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 2& 1 \\ 1 & 1 & 1 & 1& 2& \color{purple}{1}& \color{purple}{2} \\ 0 & 1 & 1 & 1& 1& \color{purple}{2}& \color{purple}{2} \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 0& 0& 0& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 2 \\ 0 & 1 & 1 & 1& 1& \color{purple}{2}& \color{purple}{2} \\ 1 & 1 & 1 & 1& 2& \color{purple}{2}& \color{purple}{1} \end{smallmatrix}}$

or also the same rules with a little $\color{purple}{\text{variation}}$ for the 7-dim. simple objects (and mult. 3 instead of 2):

$\small{ \begin{smallmatrix} 1 & 0 & 0 & 0& 0& 0& 0 \\ 0 & 1 & 0 & 0& 0& 0& 0 \\ 0 & 0 & 1 & 0& 0& 0& 0 \\ 0 & 0 & 0 & 1& 0& 0& 0 \\ 0 & 0 & 0 & 0& 1& 0& 0 \\ 0 & 0 & 0 & 0& 0& 1& 0 \\ 0 & 0 & 0 & 0& 0& 0& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 1 & 0 & 0& 0& 0& 0 \\ 1 & 1 & 0 & 1& 0& 1& 1 \\ 0 & 0 & 1 & 0& 1& 1& 1 \\ 0 & 1 & 0 & 0& 1& 1& 1 \\ 0 & 0 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 1 & 0& 0& 0& 0 \\ 0 & 0 & 1 & 0& 1& 1& 1 \\ 1 & 1 & 1 & 0& 0& 1& 1 \\ 0 & 0 & 0 & 1& 1& 1& 1 \\ 0 & 1 & 0 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 1& 0& 0& 0 \\ 0 & 1 & 0 & 0& 1& 1& 1 \\ 0 & 0 & 0 & 1& 1& 1& 1 \\ 1 & 0 & 1 & 1& 0& 1& 1 \\ 0 & 1 & 1 & 0& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 0& 1& 0& 0 \\ 0 & 0 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 0 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 0& 1& 1& 1 \\ 1 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 2& 1 \\ 0 & 1 & 1 & 1& 1& 1& 2 \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 0& 0& 1& 0 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 2& 1 \\ 1 & 1 & 1 & 1& 2& {\color{purple}{0}}& {\color{purple}{3}} \\ 0 & 1 & 1 & 1& 1& {\color{purple}{3}}& {\color{purple}{1}} \end{smallmatrix} , \begin{smallmatrix} 0 & 0 & 0 & 0& 0& 0& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 1 \\ 0 & 1 & 1 & 1& 1& 1& 2 \\ 0 & 1 & 1 & 1& 1& {\color{purple}{3}}& {\color{purple}{1}} \\ 1 & 1 & 1 & 1& 2& {\color{purple}{1}}& {\color{purple}{2}} \end{smallmatrix}}$

*Remark*: they give the *first known* non-trivial simple integral fusion rings.

These fusion rules can be found in 2 min. by SAGE with this code, giving this terminal output.

Note that $210 = 2.3.5.7$ and that these matrices are self-dual and irreducibles. They also commute.

By arXiv:0809.3031 Proposition 9.11, if such integral fusion categories exist, they couldn't be "weakly group theoretical", and by arXiv:1208.0840 Corollary 6.16, they would be abelian but not braided.

Thank you to Eric Rowell and Leonid Vainermann for these references.

Also thanks to Dave Penneys for asking Eric.

The proof that such a fusion category $\mathcal{C}$ can't be braided is the following completed argument:

If it's braided, then it can be non-degenerated (i.e. $\mathcal{C}′=Vec$) or degenerated:

- If it's non-degenerated then the contradiction follows by Corollary 6.16 cited above.

- Else it's degenerated, then $Vec \subsetneq \mathcal{C}′$, so by simplicity $\mathcal{C}′=\mathcal{C}$, so $\mathcal{C}$ is symmetric, and by Deligne (Example 4.6 here), $\mathcal{C}≃Rep(G)$ as fusion category (without considering the symmetric structure), with $G$ a finite simple group, contradiction (because there is no simple group of order $210$).

*Edit about the original motivation (July 2013)*:

These matrices are naturally came from my will of classifying the *cyclic subfactors*:

The first case we consider is "depth 2, irreducible, finite index", i.e. finite dimensional ${\rm C}^{\star}$-Hopf algebras (also called Kac algebras). The first question to answer is:

Are there non-trivial cyclic Kac algebras ? If so, the first example is certainly *maximal*.

Now a Kac algebra gives a unitary integral fusion category, so we have written this algorithm investigating all the (non-pointed) simple integral fusion rings, so in particular those related to the non-trivial maximal Kac algebras. There are finitely many possibilities for each dimension.

*Edit (June 2014)*:

We have also discovered $17$ fusion rings of FPdims $360$ and $660$, ranks $7$ and $8$, and types $[[1,1],[5,2],[8,2],[9,1],[10,1]]$ and $[[1,1],[5,2],[10,2],[11,1],[12,2]]$. Two of them come from the simple groups $A_6$ and ${PSL}(2,11)$, the $15$ others are new (see the fusion rules here).